A nonlinear elliptic curve cryptosystem based on matrices

We propose a new system that is applicable to public key cryptography. The system is a variant of the Discrete Logarithm Problem (DLP) with the elements of a certain group, formed with points of an elliptic curve, and the elements of a certain finite field related to the curve. The nonlinear term refers to the coefficient that we use as the problem to solve because it is obtained with a nonlinear combination of two scalar elements chosen at random. Also, we expose the Diffie-Hellman key agreement protocol with this system act as the underlying mathematical problem.

The system we propose is a mathematical problem with the necessary properties to define public key cryptosystems. It is based on the Elliptic Curve Discrete Logarithm Problem (ECDLP) and polynomial matrices. In this way, we achieve to increase the possible number of keys and, therefore, we augment the resolution complexity of the system. Also, we make a cryptanalisys of the system detecting its weaknesses and verifying that, even so, it is harder to solve than the ECDLP.

In this paper, discrete log-based public-key cryptography is explored. Specifically, we first examine the Discrete Log Problem over a general cyclic group and algorithms that attempt to solve it. This leads us to an investigation of the security of cryptosystems based over certain specific cyclic groups: Fp, F × p , and the cyclic subgroup generated by a point on an elliptic curve; we ultimately see the highest security comes from using E(Fp) as our group. This necessitates an introduction of elliptic curves, which is provided. Finally, we conclude with cryptographic implementation considerations.

WSEAS Transactions on …, 2008

We describe a new public key cryptosystem using block upper triangular matrices with elements in Z p , based on a generalization of the discrete logarithm problem over a finite group. The proposed cryptosystem is very efficient, requiring very few operations and also allows an ElGamal ...

International Journal of Computer Applications

In this article, a novel public key cryptosystem is introduced by using an abelian subgroup of GL(k, Z n) where n and k are positive integers. Instead of exponentiation, the conjugation automorphisms are mainly used to define the public and private keys. This allows the calculations to be fast and effective. The security analysis of the cryptosystem is discussed and it is shown that the cryptosystem is highly secure. Moreover, proposed scheme also generalizes the main scheme given in [1].

This paper studies the mathematics of elliptic curves, starting with their derivation and the proof of how points upon them form an additive abelian group. I then worked on the mathematics necessary to use these groups for cryptographic purposes, specifically results for the group formed by an elliptic curve over a finite field, E(Fq). I examine the mathematics behind the group of torsion points, to which every point in E(Fq) belongs, and prove Hasse’s theorem along with a number of other useful results. I finish by describing how to define a discrete logarithmic problem using E(Fq) and showing how this can form public key cryptographic systems for use in both encryption and decryption key exchange.

Computers, Materials & Continua

There have been many digital signature schemes were developed based on the discrete logarithm problem on a finite field. In this study, we use the elliptic curve discrete logarithm problem to build new collective signature schemes. The cryptosystem on elliptic curve allows to generate digital signatures with the same level of security as other cryptosystems but with smaller keys. To extend practical applicability and enhance the security level of the group signature protocols, we propose two new types of collective digital signature schemes based on the discrete logarithm problem on the elliptic curve: i) the collective digital signature scheme shared by several signing groups and ii) the collective digital signature scheme shared by several signing groups and several individual signers. These two new types of collective signatures have combined the advantages of group digital signatures and collective digital signatures. These signatures have a fixed size and do not depend on the number of members participating in the creation of the final collective signature. One of the advantages of the proposed collective signature protocols is that they can be deployed on top of the available public key infrastructures.

2009

In this paper, we propose a novel and efficient way to improve the computational complexity of the Elliptic Curve Cryptography [ECC] algorithm. ECC is a public key cryptography system, where the underlying calculations are performed over elliptic curves. The security of ECC is based on solving the Elliptic Curve Discrete Logarithm Problem [EDCLP]. We propose an algorithm to double the computational complexity of the conventional algorithm. The proposed algorithm generates two ECDLP opposed to one problem that was generated by the conventional algorithm being used till now. With the same key size, the proposed algorithm provides more security when compared to public key cryptography systems like RSA and ECC. It can be implemented efficiently in even less time when compared to ECC. The paper discuses the underlying protocol and proves how the enhancement in security and reduction in implementation time is achieved, thereby making it well suited for wireless communication.

— We discuss the use of elliptic curves in cryptography on high-dimensional surfaces. In particular, instead of a key exchange protocol written in the form of a bi-dimensional row, where the elements are made up with 256 bits, we propose a Diffie-Hellman key exchange protocol given in a matrix form, with four independent entries each of them constructed with 64 bits. Apart from the great advantage of significantly reducing the number of used bits, this methodology appears to be immune to attacks of the style of Western, Miller, and Adleman, and at the same time it is also able to reach the same level of security as the cryptographic system presently obtained by the Microsoft Digital Rights Management. A nonlinear differential equation (NDE) admitting the elliptic curves as a special case is also proposed. The study of the class of solutions of this NDE is in progress. Keywords— Elliptic-curve cryptography, Elliptic-curve discrete log problem, Public key cryptography, Nonlinear differential equations.

1997

The security of many cryptographic protocols depends on the di culty of solving the so-called \discrete logarithm" problem, in the multiplicative group of a nite eld. Although, in the general case, there are no polynomial time algorithms for this problem, constant improvements are being made { with the result that the use of these protocols require much larger key sizes, for a given level of security, than may be convenient.